Black-Scholes Calculator

The Black-Scholes model is the foundation of modern option pricing, and this calculator applies it to work out a fair value for a European call or put. You enter five things: the current price of the underlying share, the strike price of the option, the time left until expiry in years, the risk-free interest rate, and the volatility of the underlying. From those the calculator produces the theoretical option premium, along with the two intermediate terms d1 and d2 that drive the formula. A call option gives the right to buy at the strike, so it gains value as the share price rises above the strike; a put gives the right to sell, so it gains value as the price falls below the strike. The model rests on a set of clean assumptions: the option is exercised only at expiry, volatility and the interest rate stay constant, there are no dividends, and the share price follows a smooth random walk. Those assumptions rarely hold exactly, which is why traders adjust the volatility input to reflect market expectations, but the formula remains the standard reference point for valuing options and for understanding how each input moves the price. Change any field and the premium, d1 and d2 update immediately, so you can see how a higher volatility or a longer time to expiry lifts the value of both calls and puts.

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$
years
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$10.45
call option value (per share)
d10.350
d20.150
Put value$5.57

European exercise, no dividends, with constant volatility and interest rate. A theoretical value for education, not a trading quote. Estimate only.

How it works

The formula first standardises the position of the strike relative to the spot price. It computes d1 as the natural log of spot divided by strike, plus the rate plus half the variance, all times time, then divided by volatility times the square root of time. d2 is simply d1 minus volatility times the square root of time. The call value is the spot price times N(d1) minus the strike, discounted at the risk-free rate, times N(d2), where N is the standard normal cumulative distribution. The put value is the discounted strike times N(minus d2) minus the spot times N(minus d1). Call and put are linked by put-call parity: the call minus the put equals the spot minus the discounted strike. This calculator uses a high-accuracy polynomial approximation of the normal distribution, so the figures match published Black-Scholes tables to the cent.

Worked example

Take a share trading at $100 with a $100 strike, one year to expiry, a 5 percent risk-free rate and 20 percent volatility. Then d1 is (ln(1) plus (0.05 plus 0.02) times 1) divided by 0.20, which is 0.350, and d2 is 0.350 minus 0.20, which is 0.150. Feeding those through the normal distribution gives a call value of about $10.45 and a put value of about $5.57 per share. The two obey put-call parity, since $10.45 minus $5.57 equals $100 minus the discounted strike. These match the defaults shown in the calculator above.

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Frequently asked questions

What does the Black-Scholes model calculate?

It gives a fair theoretical price for a European call or put option, one that can only be exercised at expiry. The inputs are the current share price, the strike price, the time left to expiry, the risk-free interest rate and the volatility of the underlying. The model assumes no dividends, constant volatility and constant interest rates.

What are d1 and d2 in Black-Scholes?

They are two standardised terms the formula uses. d1 measures how far in the money the option is, adjusted for volatility and time, and N(d1) acts like the option delta. d2 equals d1 minus volatility times the square root of time, and N(d2) is the risk-neutral probability the option finishes in the money.

Does this calculator include dividends?

No. It uses the original Black-Scholes formula with no dividend yield, which suits non-dividend-paying shares and many index options. For a dividend-paying share you would lower the effective spot price or use the Merton extension, so treat the result as indicative rather than a trading quote.